Leon Ehrenpreis (1930-2010) A truly FUNDAMENTAL Mathematician (a Videotaped lecture)

By Doron Zeilberger

ל ע ל ו י   נ ש מ ת ו   ש ל   ה ר ב   א ל י ע ז ר   א ר נ פ ר י י ס   ז צ " ל

Posted: Sept. 22, 2010.

This lecture, delivered on Sept. 17, 2010 in the Rutgers University Experimental Mathematics Seminar is a mathematical eulogy to my great mentor and influencer Leon Ehrenpreis (24 Iyyar, 5690- 6 Elul, 5770) who was not only one of the greatest mathematicians of our time, and a great rabbi (see, e.g., his insightful commentary on Genesis) (and a very accomplished piano player, and a Marathon runner) but, most important, a great mensch.

You can watch the lecture either in:

Acknowledgement: The video was filmed, edited, and uploaded by Edinah Gnang
This lecture is accompanied by the Maple package LEON, that, inter alia, computes fundamental solutions to partial recurrence operators with constant coefficients, constructs natural polynomial bases to spaces of discrete functions satisfying a linear recurrence equation with constant coefficients, and finds multiplicity varieties (Ehrenpreis' brain-child) for systems of maximally over-determined systems of linear partial differential equations with constant coefficients.

# Sample Input and Output for the Maple package LEON

• In order to find the truncated generating Laurent formal power series for the symmetrized FUNDAMENTAL solution of the 2D Discete Laplacian truncated to order 20
the input gives the output.

• The table of values of the above is:
the input gives the output.

• In order to find the generating-function for the symmetrized FUNDAMENTAL solution of the 2D Duffin-Discete Laplacian,
the input gives the output.

• In order to find the table of values for the symmetrized FUNDAMENTAL solution of the 2D Duffin-Discete Laplacian,
the input gives the output.

• In order to find the generating-function for the symmetrized FUNDAMENTAL solution of the 3D Discete Laplacian, truncated to order 5 is:
the input gives the output.

• In order to see the first 10 terms of the sequence of dimensions of the space of polynomials of degree ≤ d, annihilated by the differential operators D12+D22+D3 and D13+D23+D33 , where D1, D2, D3 are differentiations w.r.t. to x,y,z resp.
the input gives the output.

• In order to see the first 10 terms of the sequence of dimensions of the space of homog. polynomials of degree d, annihilated by the differential operators D12+D22+D32 and D13+D23+D33 , where D1, D2, D3 are differentiations w.r.t. to x,y,z resp.
the input gives the output.

• In order to find the multipliticity variety for the two operators, D12-D2,D22 ,
the input gives the output.

• In order to find a basis for the polynomials annihilated by the operators D12+D22,D1+D2, of degree ≤ 2,
the input gives the output.

• In order to find a basis for the homogeneous polynomials annihilated by the operators D14+D24+D34,D13+D23+D33 of degree ≤ 4
the input gives the output.

• In order to find a basis for the polynomials in x, ..., x of degree 7, that are totally harmonic,
the input gives the output.

• In order to see a basis for the discrete analytic polynomials of degree ≤ 10,
the input gives the output.

• In order to see a basis for the discrete harmonic polynomials, in the plane, with variables m and n,
the input gives the output.

• In order to see a basis for the discrete harmonic polynomials in three dimensions, with variables m1,m2,m3
the input gives the output.

• In order to find the first ten terms of an Ehrenpreis-inspired basis for discrete analytic polynomials
the input gives the output.

• In order to find a basis for the linear vector space consisting of polynomials in m,n, of degree ≤ 5 viewed as functions on the discrete lattice with the curious property that for every 1 by 4 discrete rectangle in the plane, the value of the polynomial equals the average of its value at the other 9 locations,
the input gives the output.

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger